2D Version of Curl Integrator

[markusrenoldner]

Dear community,

what is the most straight forward way to implement LinearForm integrators and BilinearForm integrators like

mfem::VectorFEBoundaryTangentLFIntegrator()
mfem::MixedCrossProductIntegrator()

for 2D meshes?

[tzanio]

ping: @mlstowell

[mlstowell]

Hi, @markusrenoldner,

The 2D version of mfem::MixedCrossProductIntegrator() is the mfem::MixedScalarCrossProductIntegrator(). If that isn't exactly what you need you may be able to modify it to suit your purposes. Note that related integrators mfem::MixedScalarCurlIntegrator() and mfem::MixedScalarWeakCurlIntegrator() may also be of interest.

I'm afraid we don't yet have any detailed guidance on writing your own integrators. However, I'd be happy to offer advice. Specifically which operators/integrators do you need?

Best wishes,
Mark

[markusrenoldner]

Hello!

I see, so what I am trying to implement is a 2D version of this BilinearForm:
$$\left\langle u \times v, w\right\rangle \qquad \forall w$$

which corresponds to the MixedCrossProductIntegrator. Moreover, I believe that the boundary integral
$$(u\times n, w)_{\partial \Omega}\qquad \forall w$$

implemented by VectorFEBoundaryTangentLFIntegrator also only works in 3D and I am looking for a 2D replica as well.

Interestingly, i found that MixedVectorCurlIntegrator seems to work in 2D, this I did not expect due to the curl operator.

[mlstowell]

Hi, @markusrenoldner,

Just to be clear, what spaces are u and w meant to be in?

Best wishes,
Mark

[markusrenoldner]

Hello!
I am actually interested in both $u, v, w\in H(curl)$ as well as $u, v, w\in H(div)$ for the MixedCrossProductIntegrator. (In the discrete setting these terms should all be square integrable).

For the VectorFEBoundaryTangentLFIntegrator, I have $u\in H(div)$ and $w\in H(curl)$.

[mlstowell]

Hi, @markusrenoldner,

I'm afraid what you're asking for is inconsistent. If u and v are in HCurl in 2D then their cross product will be a scalar so w cannot be in HCurl. On the other hand u and w could be in HCurl if v were scalar and interpreted as a vector perpendicular to the 2D surface. Unless you're planning to use the ND_R2D finite elements which can represent 3 component vector fields in 2D surfaces. If the later is your intention then this will require special attention since many of our bilinear integrators were written before these ND_R2D basis functions were implemented.

Best wishes,
Mark

[markusrenoldner]

Good morning,

you are right, I would require $v$ (or $u$) to be a vector perpendicular to the 2D domain. It sounds as if this would be quite some work to implement, which I dont have time for at the moment.

Still, thank you for the response!